The distribution of patterns in random trees

Authors:
FrÉdÉric Chyzak;Michael Drmota;Thomas Klausner;Gerard Kok
Affiliations:
Inria-rocquencourt, f-78153 le chesnay cedex, france (e-mail: frederic.chyzak@inria.fr);Institut für diskrete mathematik und geometrie, technische universität wien, wiedner hauptstraße 8-10/113, a-1040 wien, austria (e-mail: michael.drmota@tuwien.ac.at, klausner@dmg.tu ...;Institut für diskrete mathematik und geometrie, technische universität wien, wiedner hauptstraße 8-10/113, a-1040 wien, austria (e-mail: michael.drmota@tuwien.ac.at, klausner@dmg.tu ...;Delft institute of applied mathematics, delft university of technology, mekelweg 4, nl-2628 cd delft, the netherlands (e-mail: gkok@fsmat.at)
Venue:
Combinatorics, Probability and Computing
Year:
2008

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Abstract

Let 𝒯n denote the set of unrootedlabelled trees of size n and let m be aparticular (finite, unlabelled) tree. Assuming that every tree of𝒯n is equally likely, it is shown thatthe limiting distribution as n goes to infinity of thenumber of occurrences of m is asymptotically normalwith mean value and variance asymptotically equivalent toμn and σ2n, respectively, wherethe constants μ0 and σ≥0 are computable.